BuilderGrade 7⏱ 20 min

Circles and π

Roll a wheel to unroll its circumference, discover π, then slice a circle into a near-rectangle to find its area.

Standards: 7.G.B.4

1Wonder

Take any round thing (a coin, a plate, a bicycle wheel) and wrap a string around it. The string is always a little more than 3 times as long as the circle is wide. Always. Why that number, and what is it exactly?

2Explore

Circle lab

This activity needs a bigger screen

Turn your phone sideways, or open this lesson on a tablet or computer. The explanation and quiz below work here.

What to explore

Diameter d 3

Try this

Press Roll one turn. How many diameters fit along the red line?
Change the diameter and roll again. Does C ÷ d change?
Switch to Inside: area and turn on Rearrange the slices. What shape do the slices make?
Use more slices. Why does the shape look more and more like a rectangle?

3Explain

Step 1 of 5

The parts of a circle

The radius r goes from the center to the edge. The diameter d goes all the way across, so d = 2r. The distance around is the circumference C.

Circles

\begin{aligned} C &= \pi d = 2\pi r \\ A &= \pi r^2 \\ \pi &\approx 3.14159\ldots \end{aligned}

For every circle, the circumference is π times the diameter. Slicing a circle and rearranging the pieces shows why its area is π r².

4Practice

Question 1 of 5

A circle has a radius of 5 cm. What is its diameter?

5Recap

d = 2r; the distance around is the circumference.
C ÷ d is the same for every circle: π ≈ 3.14159…
C = πd = 2πr.
A = πr²: slice the circle and rearrange it into a near-rectangle πr by r.